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Energetic Variational Approaches for inviscid multiphase flow systems with surface flow and tension

Hajime Koba

Abstract

We consider the governing equations for the motion of the inviscid fluids in two moving domains and an evolving surface from an energetic point of view. We employ our energetic variational approaches to derive inviscid multiphase flow systems with surface flow and tension. More precisely, we calculate the variation of the flow maps to the action integral for our model to derive both surface flow and tension. We also study the conservation and energy laws of our multiphase flow systems. The key idea of deriving the pressure of the compressible fluid on the surface is to make use of the feature of the barotropic fluid, and the key idea of deriving the pressure of the incompressible fluid on the surface is to apply a generalized Helmholtz-Weyl decomposition on a closed surface.

Energetic Variational Approaches for inviscid multiphase flow systems with surface flow and tension

Abstract

We consider the governing equations for the motion of the inviscid fluids in two moving domains and an evolving surface from an energetic point of view. We employ our energetic variational approaches to derive inviscid multiphase flow systems with surface flow and tension. More precisely, we calculate the variation of the flow maps to the action integral for our model to derive both surface flow and tension. We also study the conservation and energy laws of our multiphase flow systems. The key idea of deriving the pressure of the compressible fluid on the surface is to make use of the feature of the barotropic fluid, and the key idea of deriving the pressure of the incompressible fluid on the surface is to apply a generalized Helmholtz-Weyl decomposition on a closed surface.
Paper Structure (10 sections, 11 theorems, 155 equations, 1 figure)

This paper contains 10 sections, 11 theorems, 155 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Variations of flow maps and main results
  3. Preliminaries
  4. Variations of action integral with respect to flow maps
  5. Mathematical modeling
  6. Inviscid multiphase flow system with compressible surface flow
  7. Inviscid multiphase flow system with incompressible surface flow
  8. Inviscid multiphase flow system with tangential surface flow
  9. Conservation and energy laws
  10. Appendix

Key Result

Proposition 2.7

The following two assertions hold: $(\rm{i})$ Assume that for every $0< t <T$ and $\Lambda \subset \Omega$, Then $(\rho_A , \rho_B , \rho_S )$ satisfies $(\rm{ii})$ Let $-1< \varepsilon < 1$. Assume that for every $0< t <T$ and $\Lambda \subset \Omega$, Then $(\rho^\varepsilon_A , \rho^\varepsilon_B , \rho^\varepsilon_S )$ satisfies Here $D_t^{A,\varepsilon} f := \partial_t f + (v^\varepsilon_

Figures (1)

  • Figure 1: Moving Domains and Surfaces

Theorems & Definitions (26)

  • Remark 1.1
  • Remark 1.2
  • Definition 2.1: Moving domains and surface
  • Definition 2.2: Function spaces in moving domains and surfaces
  • Definition 2.3: Flow maps in $\overline{\Omega_T}$
  • Definition 2.4: Variations of domains and surface
  • Definition 2.5: Flow maps in $(\overline{\Omega^\varepsilon_{A,T}} , \overline{\Omega^\varepsilon_{B,T}} , \overline{\Gamma^\varepsilon_T})$
  • Definition 2.6: Variations of flow maps in $(\overline{\Omega^\varepsilon_{A,T}} , \overline{\Omega^\varepsilon_{B,T}} , \overline{\Gamma^\varepsilon_T})$
  • Proposition 2.7: Continuity equations
  • Theorem 2.8: Variations of the flow maps to the action integral
  • ...and 16 more